二层次消费与投资决策优化动态规划模型

36320063MATHEMATICSINPRACTICEANDTHEORYVol136No13March,20061,2,3(1.,350002)(2.,344000)(3.,200051):,,..,,,,.,,..:;;;-:2004203229:(2003034280);(0511006)1HarryMarkowitz[1](1952),,,.Markowitz-,WilliamF.Sharpe[2](1964)JohnLint2ner[3](1965)JohnMoosin[4](1966),,Markowitz-,(CAPM).CAPM,,.CAPM:CAPM,.,7080,D.Breeden[5]M.Rubinstein[6]EdwinJ.EltonM.J.Gruber[7],CAPM,CAPM(CCAPM).,F.Fama[8]CAPM.R.C.Merton[9]CAPM(ICAPM).JohnC.CoxJonathonE.IngersollStephenA.Ross[10]ICAPM.,,.MertonCox,,,[11].,,.,,,,.,,.2t,t+1,,T,,U(õ),U1(õ).n+1,,n.s(tsT),Rf(s)=1+rf(s),rf(s);iPi(s),Ri(s)=Pi(s)öPi(s-1)=1+rui(s),rui(s),i=1,2,,n.R(s)=(R1(s),R2(s),,Rn(s)),Ri(s)=ERi(s),R(s)=(R1(s),R2(s),,Rn(s)),Var(R(s))=(s)(R(s)),.sY(s),C(s),sW(s),(:,.)sI(s)=W(s-1)+Y(s)-C(s),ixi(s)=Ni(s)Pi(s)öI(s),Ni(s)i.x(s)=(x1(s),x2(s),,xn(s)),l=(1,1,,1),x0(s)=1-lx(s).MertonCox,,:,.maxEt6Ts=tU(C(s),s)+U1(W(T),T)(1)s.t.W(s)=[W(s-1)+Y(s)-C(s)][x(s)R(s)+x0(s)Rf(s)]x0(s)+lx(s)=1s=t,t+1,,TW(t-1)=W0.,,,,,,.,,.,..:C(t),,Markowitz-,(),C(t),.max[U(C(t),t)+U1(W(t),t)](2a)s.t.W(t)=max[(W(t-1)+Y(t)-C(t))(x(t)(R(t)-Rf(t)l)+Rf(t))](2b)s.t.x(t)(t)x(t)=R2p(t),W(t-1)=W01C(t),,Markowitz-,(),,C(t),6436Rp(t)=H(t)Rp(t)+Rf(t),H2(t)=(R(t)-Rf(t)l)-1(t)(R(t)-Rf(t)l),U(C(t),t)+U1(W(t),t).(2b).Lagrange(),L=I(t)(x(t)(R(t)-Rf(t)l)+Rf(t))+K(x(t)(t)x(t)-R2p(t))Lx(t)=I(t)(R(t)-Rf(t)l)+2K(t)x(t)=0(3a)LK=x(t)(t)x(t)-R2p(t)=0(3b)(3a),xp(t)=-I(t)-1(t)(R(t)-Rf(t)l)ö2K(4)(4)(3b),I2(t)(R(t)-Rf(t)l)-1(t)(R(t)-Rf(t)l)=4K2R2p(t)(5)(5),K=I(t)H(t)ö(2Rp(t))(6),H2(t)=({R(t)-Rf(t)l)-1(t)(R(t)-Rf(t)l)(7)(6)(4),xp(t)=Rp(t)-1(t)(R(t)-Rf(t)l)öH(t)(8)(xp(t)=Rp(t)-1(t)(R(t)-Rf(t)l)öH(t),.)(8)(2b),(7),C(t)W(t)=(W(t-1)+Y(t)-C(t))(H(t)Rp(t)+Rf(t))=(W(t-1)+Y(t)-C(t))Rp(t)(9),Rp(t)=H(t)Rp(t)+Rf(t),.(2a),(9).dUdC(t)+dU1dW(t)dW(t)dC(t)=0(10)dW(t)dC(t)=-Rp(t),dUdC(t)=Rp(t)dU1dW(t)dUdC(t)ödU1dW(t)=Rp(t)(11)U(õ),U1(õ),C(t)d2UdC(t)2+R2p(t)d2U1dW(t)20(11),C3(t),W3(t)=(W(t-1)+Y(t)-C3(t))Rp(t).,C(t),C3(t),Rp(t),..1743,:U(C(t),t)+U1(W(t),t)=C(t)-aC2(t)ö2+W(t)-bW2(t)ö2(12)a0,b0,.(11)(12)1-aC(t)=Rp(t)(1-bW(t))(13)(9)(13),C3(t)=(1+(W0+Y(t))R2p(t)-Rp(t))ö(a+bR2p(t))(14a)W3(t)=(W0+Y(t)-C3(t))Rp(t)(14b)Rp(t)=H(t)Rp(t)+Rf(t)(14c),,,().3,:,Markowitz-,,.max6Ts=tU(C(s),s)+U1(W(T),T)(15a)s.t.W(s)=max[(W(s-1)+Y(s)-C(s))(x(s)(R(s)-Rf(s)l)+Rf(s))](15b)s.t.x(s)(s)x(s)=R2p(s)s=t,t+1,,T,W(t-1)=W02C(s),,Markowitz-,(),,C(s),s(T)G(s)=7Tk=sRp(k),s=t,t+1,,T,6Ts=tU(C(s),s)+U1(W(T),T),Rp(k)=H(k)Rp(k)+Rf(k).(15b).1,xp(s)=Rp(s)-1(s)(R(s)-Rf(s)l)öH(s)(16a)W(s)=(W(s-1)+Y(s)-C(s))Rp(s)(16b)s=t,t+1,,T,Rp(s)=H(s)Rp(s)+Rf(s),H2(s)=(R(s)-Rf(s)l)-1(s)(R(s)-Rf(s)l),.(15a),(16b)..T,:C(T),J(W(T-1),T)max[U(C(T),T)+U1(W(T),T)](17)8436s.t.W(T)=(W(T-1)+Y(T)-C(T))Rp(T)dUdC(T)+dU1dW(T)dW(T)dC(T)=dUdC(T)-Rp(T)dU1dW(T)=0dUdC(T)=Rp(T)dU1dW(T)dUdC(T)ödU1dW(T)=Rp(T)(18)U(õ),U1(õ),(18),C3(T).J(W(T-1),T)=U(C3(T),T)+U1(W(T),T)(19a)W(T)=(W(T-1)+Y(T)-C3(T))Rp(T)(19b)T-1,,J(W(T-2),T-1)max[U(C(T),T)+U(C(T-1),T-1)+U1(W(T),T)]=max[U(C(T-1),T-1)+J(W(T-1),T)](20)s.t.W(T-1)=(W(T-2)+Y(T-1)-C(T-1))Rp(T-1)dUdC(T-1)+dJdW(T-1)dW(T-1)dC(T-1)=dUdC(T-1)-Rp(T)Rp(T-1)dU1dW(T)=0dUdC(T-1)=Rp(T)Rp(T-1)dU1dW(T)dUdC(T-1)ödU1dW(T)=Rp(T)Rp(T-1)(21)U(õ),U1(õ),(21),C3(T-1).J(W(T-2),T-1)=(C3(T-1),T-1)+U(C3(T),T)+U1(W(T),T)(22a)W(T)=(W(T-2)+Y(T-1)-C3(T-1))Rp(T)Rp(T-1)+(Y(T)-C3(T))Rp(T)(22b),dUdC(s)ödU1dW(T)=G(s),s=t,t+1,,T(23)G(s)=7Tk=sRp(k).C3(s),s=t,t+1,,T,J(W(s-1),s)=6Tk=sU(C3(k),k)+U1(W(T),T)(24a)W(T)=W(s-1)G(s)+6Tk=s(Y(k)-C3(k))G(k)(25b)s=t,t+1,,T(2a)J(W0,t)=6Ts=tU(C3(s),s)+U1(W3(T),T)(26a)W3(T)=W0G(t)+6Ts=t(Y(s)-C3(s))G(s)(26b)943,:,C(s),sTG(s)=7Tk=sRp(k),s=t,t+1,,T,6Ts=tU(C(s),s)+U1(W(T),T)...,,.,,(23),.,,,..2U(C(s),s)=(s)-aC2(s)ö2,U1(W(T),T)=W(T)-bW2(T)ö2(27)s=t,t+1,,T,a0,b0,C3(s),s=t,t+1,,TW3(T).(23)(27)1-aC3(s)=G(s)(1-bW3(T)),s=t,t+1,,T(28)a(Y(s)-C3(s))G(s)=G2(s)(1-bW3(T))+(aY(s)-1)G(s),s=t,t+1,,T(29),(26b),a(W3(T)-W0G(t))=6Ts=t[G2(s)(1-bW3(T))+(aY(s)-1)G(s)](30)(30),(28),W3(T)=[6s=tT(G(s)+aY(s)-1)G(s)+aW0G(t)]ö[a+b6Ts=tG2(s)](31a)C3(s)=[1+(bW3(T)-1)G(s)]öa,s=t,t+1,,T(31b)G(s)=7Tk=sRp(k),Rp(k)=H(k)Rp(k)+Rf(k).,,,().,(1),(15),,,,.[12],,,,,,,,.,,..0536:[1]MarkowitzHM.Portfolioselection[J].JournalofFinance,1952,(7):7791.[2]SharpeWF.Capitalassetprices:atheoryofmarketequilibriumunderconditionsofrisk[J].JournalofFinance,1964,(9):425442.[3]LintnerJ.Securityprices,risk,andmaximalgainsfromdiversification[J].JournalofFinance,1965,(10):587615.[4]MossinJ.Equilibriuminacapitalassetmarket[J].Econometrica,1966,(10):768783.[5]BreedenD.Anintertemporalassetpricingmodelwithstochasticconsumptionandinvestmentopportunities[J].JournalofFinancialEconomics,1979,(7):265296.[6]Rubinstein.Thevaluationofuncertainincomestreamsandthepricingofoptions[J].BellJournalofEconomicsandManagementScience,1976,(7):407435.[7]EltonEJ,GruberMJ.Taxesandportfoliocomposition[J].1978,(6):399410.[8]FamaEF.Efficientcapitalmarket:areviewoftheoryandempiricalwork[J].JournalofFinance,1970,25(2):383417.[9]MertonR.Theoryofrationaloptionpricing[J].BellJournalofEconomicsandManagementScience,1973,(4):141183.[10]CoxJC.IngersollJE,RossSA.Anintertemporalgeneraleq